A292870 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of k-th power of continued fraction 1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))).

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 5, 0, 1, 4, 9, 14, 15, 0, 1, 5, 14, 28, 44, 52, 0, 1, 6, 20, 48, 93, 154, 203, 0, 1, 7, 27, 75, 169, 333, 595, 877, 0, 1, 8, 35, 110, 280, 624, 1289, 2518, 4140, 0, 1, 9, 44, 154, 435, 1071, 2442, 5394, 11591, 21147, 0, 1, 10, 54, 208, 644, 1728, 4265, 10188, 24366, 57672, 115975, 0

Offset: 0

Author

Ilya Gutkovskiy, Sep 25 2017

Keywords

• nonn
• tabl

Comments

A(n,k) is the n-th term of the k-fold convolution of Bell numbers with themselves. - Alois P. Heinz, Feb 12 2019

Examples

			G.f. of column k: A_k(x) = 1 + k*x + k*(k + 3)*x^2/2 + k*(k^2 + 9*k + 20)*x^3/6 + k*(k^3 + 18*k^2 + 107*k + 234)*x^4/24 + k*(k^4 + 30*k^3 + 335*k^2 + 1770*k + 4104)*x^5/120 + ...
Square array begins:
  1,   1,    1,    1,    1,     1,  ...
  0,   1,    2,    3,    4,     5,  ...
  0,   2,    5,    9,   14,    20,  ...
  0,   5,   14,   28,   48,    75,  ...
  0,  15,   44,   93,  169,   280,  ...
  0,  52,  154,  333,  624,  1071,  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened

Crossrefs

Columns k=0-4 give A000007, A000110, A014322, A014323, A014325.

Rows n=0-3 give A000012, A001477, A000096, A005586.

Antidiagonal sums give A137551.

Main diagonal gives A292871.

Cf. A205574 (another version).

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
     `if`(k=1, add(A(n-j, k)*binomial(n-1, j-1), j=1..n),
     (h-> add(A(j, h)*A(n-j, k-h), j=0..n))(iquo(k,2)))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 31 2018

Mathematica

Table[Function[k, SeriesCoefficient[1/(1 - x + ContinuedFractionK[-i x^2, 1 - (i + 1) x, {i, 1, n}])^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

G.f. of column k: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^k, a continued fraction.